In this article we study the validity of the Whitney $$C^1$$ C 1 extension property for horizontal curves in sub-Riemannian manifolds that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a suitable non-singularity property holds for the endpoint map on the Carnot groups obtained by nilpotent approximation. We then discuss the case of sub-Riemannian manifolds with singular points and we show that all step-2 manifolds satisfy the $$C^1$$ C 1 extension property. We conclude by showing that the $$C^1$$ C 1 extension property implies a Lusin-like approximation theorem for horizontal curves on sub-Riemannian manifolds.
Calculus of Variations and Partial Differential Equations – Springer Journals
Published: Mar 12, 2018
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